Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding

He, Haoze (avtor), Kressner, Daniel (avtor), Plestenjak, Bor (avtor)

It is well known that a family of $n \times n$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the $n$ joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to such joint eigenvalues for a family of (nearly) commuting matrices. This task arises, for example, in solvers for multiparameter eigenvalue problems and systems of multivariate polynomials, which are our main motivations. We propose and analyze a simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family. We provide some analysis and numerous numerical examples, showing that such randomized approaches can compute semisimple joint eigenvalues accurately and lead to improved performance of existing solvers.

• Jezik: Angleški jezik
• Ključne besede: commuting matrices, joint eigenvalues, Rayleigh quotient, randomized numerical linear algebra, multiparameter eigenvalue problems, polynomial systems
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Leto izida: 2025
• Št. strani: Str. 861–892
• Številčenje: Vol. 100, iss. 3
• PID: 20.500.12556/RUL-175108
• UDK: 519.6
• ISSN pri članku: 1017-1398
• DOI: 10.1007/s11075-024-01971-0
• COBISS.SI-ID: 215046403
• Datum objave v RUL: 16.10.2025

Gradivo je del revije

• Naslov: Numerical algorithms
• Skrajšan naslov: Numer. algorithms
• Založnik: Springer Nature
• ISSN: 1017-1398
• COBISS.SI-ID: 30694400

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.

Sekundarni jezik

• Jezik: Slovenski jezik
• Ključne besede: komutirajoče matrike, skupne lastne vrednosti, Rayleighov kvocient, verjetnostna numerična linearna algebra, večparametrični problemi lastnih vrednosti, polinomski sistemi